平移量子仿射超代数
Shifted quantum affine superalgebras
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中文总结 AI 辅助
本文为量子仿射超代数引入平移版本，研究标准奇偶型A代数的三角分解、PBW--Drinfeld基、RTT实现及Ding--Frenkel同构，并构造余结合余积。
中文摘要 AI 辅助
对于具有固定Drinfeld电流表示的量子仿射超代数，我们通过修改Cartan--Drinfeld电流的首项次数，同时保留其余电流关系和Serre关系，定义了其平移版本。随后，我们研究标准奇偶型A代数$\mathcal U_q^{\mu_+,\mu_-}(\widehat{\mathfrak{sl}}_{m|n})$。对于任意平移以及$m,n\ge1$且$m\ne n$的情况，我们证明了三角分解的存在性。当$m,n>1$时，我们获得了任意平移下的PBW--Drinfeld基，以及对于每个反支配的$\mu$，单侧代数$\mathcal U_q^{0,\mu}$的Levendorskii型有限表示。对于反支配的双侧平移，我们构造了平移RTT实现，并证明了相应的Ding--Frenkel同构。我们构造了任意双侧平移下的平移形式Drinfeld余积，该余积在完备张量积中是余结合的。平移RTT实现在反支配区域产生非完备的单侧余积，并通过有限表示在有限多个生成元上明确给出；单射平移同态将其扩展到任意单侧平移。当中间平移为反支配时（包括边界情况），其三重余结合性成立。最后，对于双侧反支配平移$\mu_+,\mu_-\in\Lambda^-$，平移的全一般线性Ding--Frenkel同构将特殊线性单电流代数识别为在公共标量重标度下的固定代数，并产生余结合的双侧非完备余积。
英文摘要
For a quantum affine superalgebra with a fixed Drinfeld current presentation, we study shifted quantum affine superalgebras through a two-sided shifted Drinfeld current presentation, in which the leading degrees of the Cartan--Drinfeld currents are modified while the remaining current and Serre relations are retained. We then focus on the standard-parity type A algebras $\mathcal U_q^{μ_+,μ_-}(\widehat{\mathfrak{sl}}_{m|n})$. For arbitrary shifts and $m,n\ge1$, $m\ne n$, we prove a triangular decomposition. When $m,n>1$, we obtain PBW--Drinfeld bases for arbitrary shifts and a Levendorskii-type finite presentation of the one-sided algebra $\mathcal U_q^{0,μ}$ for every antidominant $μ$. For antidominant two-sided shifts, we construct shifted RTT realizations and prove the corresponding Ding--Frenkel isomorphisms. We construct a shifted formal Drinfeld coproduct for arbitrary two-sided shifts, coassociative in a completed tensor product. Shifted RTT realizations yield non-completed one-sided coproducts in the antidominant region, made explicit on finitely many generators by the finite presentation; injective shift homomorphisms extend them to arbitrary one-sided shifts. Their ternary coassociativity holds whenever the middle shift is antidominant, including boundary cases. Finally, for two-sided antidominant shifts $μ_+,μ_-\inΛ^-$, a shifted full general-linear Ding--Frenkel isomorphism identifies the special-linear simple-current algebra as the fixed algebra under common scalar rescalings and yields coassociative two-sided non-completed coproducts.
发表机构
- School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）
- Faculty of Mathematics, University of Seville（塞维利亚大学数学系）
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